Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Romanian Reports in Physics, Vol. 64, No. 1, P. 294–301, 2012 ON THE PROBLEM OF THE TWO-BODIES OF VARIABLE MASS SIMION BOGÎLDEA High School “Regele Ferdinand”, Sighetu Marmatiei, Romania E-mail address: [email protected] Received October 15, 2011 Abstract. In this paper we intend to present an analytical solution of the problem which refers to the increase of the major semi-axis of the planetary orbits in the case of the two-bodies in the secondorder of approximation. The obtained results are compared with those reported by Vescan [9] for the first-order approximation and with those obtained numerically from the established basic equations. Key words: gravitation-stars: variable mass-mass loss-celestial mechanics-two body problem; methods: analytical and numerical results. 1. INTRODUCTION Many studies were done in the 20th century on celestial mechanics which lead to systems of differential equations that model the two-body problem in the case of variable mass. An example is the problem studied by the Romanian physicist Vescan [9] who studied in his PhD thesis the influence of variation of solar mass (as a result of solar radiation) about the major semi-axis of the planetary orbits. This interesting problem is an astronomical application of the two-body problem in the case of variable mass, a problem which was studied for the first time by the Swedish astronomer Gylden [1]. Problems connected by the motion of celestial bodies with variable mass, which refer to the decrease of the mass as a result of corpuscular star radiation have had a great importance in the last several decades. In these studies it is usually considered that the law of mass variation of a star (particularly of the Sun) is of the following form , (1) where α is a constant and n is a positive real number. We notice that the case n = 1 corresponds to the law of mass variation used by Vescan [9] and the case n = 2 was studied by Meshcherskii [5]. Many papers which have studied the laws 2 On the problem of the two-bodies of variable mass 295 of mass variation of the central body and the problem of the two-bodies with decreasing mass can be found in the literature, such as, for example, by Jeans [4], Hadjidemetriou [2, 3], Schröder and Smith [8]. Some refined analytical solutions are given in the paper Pal et al. [7]. In this article we will use the linear expression of the mass variation to establish the differential equation of the trajectory, the form of planetary orbits and the increase of the major semi-axis in the approximation of the second-order. The obtained analytical solution is compared with the results reported by Vescan [9] and with those resulted from the integration of equations using numerical methods. The obtained results show a very good agreement between them. 2. THE BASIC EQUATIONS By analogy with the radioactivity, Vescan [9] has considered the law of variation of mass solar of the form , (2) where q is a very small constant, t is the time and µ0 is the mass of the Sun at the initial time (initial moment) t0 . By development in series and limiting to the first two terms, it results in the following linear expression . (3) Then, the Newton’s law of attraction between the Sun and the planet of mass m can be written as . (4) In order to obtain the equation of the planetary orbit, Vescan [9] has used Picard’s successive approximation method and obtained the following equation of the trajectory . (5) This equation shows that the trajectory of one planet has the form of an elliptical spiral or an increasing rosette. The increase of the major semi-axis during the revolution ∆aV obtained by Vescan [9] is . (6) 296 Simion Bogîldea 3 3. DIFFERENTIAL EQUATION OF THE TRAJECTORY Because the attraction force is a central one, we have , (7) where C is the constant area which has the expression , (8) Then, the Newton’s law (4) becomes ], (9) where (10) is an even function. The expression of r 2 (θ) developed in a Fourier trigonometrically series is given by (11) ]. For the approximation of the second order ( n = 2 ), we have , , . (12) On the other hand, from (9) we get . (13) Using Binet’s formula (14) together with the equations (9) and (13) it obtains 4 On the problem of the two-bodies of variable mass 297 , (15) and it represents the differential equation of trajectory. 4. THE FORM OF THE PLANETARY ORBITS AND THE EXTENSION OF THE MAJOR SEMI-AXIS In order to solve equation (15) we look for a solution of the form (16) where ψ (θ) is a correction factor. Using (15) and (16) it is found that the function ψ (θ) must satisfy the ordinary differential equation (17) , where the prime denotes differentiation with respect to θ . Further, using the form of equation (17), we assume that its solution is given by , where M, values for and and (18) are unknown functions of having very small numerical . Substituting (18) into (17), we get, by the identification of the following algebraic system M + β = 1, , β + 2γθ = 0, . The solution of this system is M = 1 β. (19) 298 Simion Bogîldea 5 . (20) Thus (18) becomes . (21) From (16) and (21) we get the form of the planetary orbits: . (22) From the expression (22) it can be seen that one more term appears besides the result obtained by Vescan [9] in his equation (5). Thus, the extension of the major semi-axis during a revolution is given by . (23) 5. NUMERICAL RESULTS On the other side, we determine numerically the quantity , (24) using the following system (see Muchametkalieva and Omarov [6]; Hadjidemetriou, J. [2, 3]) ; ; ; 6 On the problem of the two-bodies of variable mass 299 , (25) where a, e, ω, θ and µ represent the major semi-axis, the eccentricity, the argument of the perihelion, the true anomaly and the mass of the Sun. The system (25) was integrated numerically for a revolution using a Fehlberg fourth-fifth order Runge-Kutta method (MAPLE, rkf45) for the values of the quantities q , K and µ0 considered by Vescan [9] . (26) The initial values (at the time t = 0 ) of the parameters a, e, ω and θ are given in Table 1. 6. CONCLUSIONS The aim of this paper is to determine analytically the expressions of ∆a1 given by (23) and numerically the expression of ∆a . The results are presented in Table 2 where there are included also the analytical results obtained by Vescan [9] given by Eqs. (6). Comparing the results given in this table, obtained using the analytical and numerical solutions it can be concluded that there is a good agreement between them. Therefore, we are confident that the system (25) can be used also for the present problem. On the other hand, from the numerical integration of the system (25) it has been found that the variation of the eccentricity is oscillating on time. But, the amplitude of the oscillation is very small (taking as on example the Earth the difference between the maximum and minimum for ten years is of order 10 −14 and for one thousand years it is 10 −11 ), that means that the variation of the eccentricity is very small. It means that the eccentricity is constant. This results is in very good agreement with the that obtained by Jeans [4]. Table 1 Values of a0 , e0 , ω0 and θ0 at the initial time, t = 0 Planet (rad/s) (cm) Mercury Venus 12 (rad) 5.8·10 0.205631752 0 1.08·1013 0.006771882 0 300 Simion Bogîldea 7 Table 1 (continued) 13 Earth 1.5·10 0.016708617 0 Mars 2.3·1013 0.09340062 0 Jupiter 7.8·1013 0.048494851 0 Saturn 1.4·1014 0.055508622 0 Uranus 2.9·1014 0.046295899 0 Neptune 4.5·1014 0.008988095 0 Pluto 6.0·1014 0.248 0 Table 2 Planet ∆av Eq. (6) (cm) ∆a1 Eq. (23) (cm) ∆a Present numerical results (cm) Mercury 0.2206957839 0.2324573618 0.22168 Venus 1.044199404 1.045353061 1.0488 Earth 2.373844135 2.384145714 2.3690 Mars 6.911043753 7.077387973 6.8046 Jupiter 146.3728602 148.2003297 145.9688 Saturn 631.7483878 640.7767262 652.952 Uranus 3901.387136 3947.887234 3838.938 Neptune 11701.87184 11729.51855 11701.188 Pluto 24021.61369 25570.68070 23075.000 Acknowledgements. The author wish to express his gratitude to professor Ioan M. Pop and dr. Tiberiu Oproiu for useful comments and references. REFERENCES 1. Gylden H., Astron. Nachr., Bd.109, 1884 pp.1–6. 2. Hadjidemetriou, J., Two-body problem with variable mass: a new approach, Icarus, 2, 404 (1963). 8 On the problem of the two-bodies of variable mass 301 3. Hadjidemetriou, J., Analytic solutions of the two-body problem with variable mass, Icarus, 5, 34–46 (1966). 4. Jeans, J.H., Astronomy and Cosmogony, Dover Publications, New York, 1961, pp. 298–301 (Reprinted from Cambridge University Press, 2nd Edition, 1929). 5. Meshcherskii, F., Mechanics of variable mass (in Russian), Moskow, 1952. 6. Muchametkalieva, R.K. and Omarov, T.B., Proceedings of Astrophysics Institute (in Russian), Moskow, Vol. XXVIII, 1976. 7. Pal, A.,Şelaru, D.,Mioc, V.,Cucu-Dumitrescu, C., The Gyldén-type problem revisited: more refined analytical solutions, Astron.Nachr., 327, 304–312 (2006). 8. Schröder, K.P. and Robert Connon Smith, Distant future of the Sun and Earth revisited, Mon. Not. R. Astron. Soc. 000, 1–10 (2008). 9. Vescan, T., Contribuţiuni la teoria cinetică şi relativistă a fluidelor reale, PhD thesis,Cluj, 1939. Anuarul Astronomic 1999, Editura Academiei Române, Bucharest, 1999. 10.